European Journal of Operational Research 240 (2015) 781–790

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

Production-driven opportunistic maintenance for batch productionbased on MAM–APB scheduling

http://dx.doi.org/10.1016/j.ejor.2014.08.0040377-2217/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: State Key Laboratory of Mechanical System andVibration, Department of Industrial Engineering, School of Mechanical Engineering,Shanghai Jiao Tong University, Shanghai 200240, China. Tel./fax: +86 21 3420 6539.

E-mail addresses: xtbxtb@sjtu.edu.cn, tangbinxia1121@gmail.com (T. Xia).

Tangbin Xia a,b,⇑, Xiaoning Jin b, Lifeng Xi a, Jun Ni b

a State Key Laboratory of Mechanical System and Vibration, Department of Industrial Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University,Shanghai 200240, Chinab Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2136, USA

a r t i c l e i n f o

Article history:Received 10 January 2014Accepted 2 August 2014Available online 11 August 2014

Keywords:MaintenanceBatch productionAdvance-postpone balancingSet-up opportunityCost saving

a b s t r a c t

Nowadays, modern production patterns, such as batch production, have brought new challenges formulti-unit maintenance decision-making. The maintenance scheduling should not only consider individ-ual machine deterioration, but also apply to batch production with variable lot size. An interactivebi-level maintenance strategy is thus proposed in a multi-unit batch production system with degradingmachines. In the machine-level scheduling, a multi-attribute model (MAM) is used to obtain mainte-nance intervals according to individual machine degradation. In the system-level scheduling, a novelproduction-driven opportunistic maintenance strategy is developed by considering both machinedegradation and characteristics of batch production. In this strategy, advance-postpone balancing(APB) utilizes set-up times as opportunities to make real-time schedules for system-level maintenance.The numerical example shows that the proposed MAM–APB methodology can efficiently eliminateunnecessary production breaks, achieve significant cost reduction and overcome complexity of systemscheduling.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

In today’s competitive industrial market, maintenance schedul-ing has played a more and more important role to meet the inno-vation requirements of modern production patterns. However,huge cost wastes are caused by improper scheduling in realisticenvironments (Mobley, 2002; Ruiz-Castro & Li, 2011). To satisfythe increasing customer demands, multi-unit manufacturing sys-tems need appropriate maintenance policy to achieve high systemperformance with minimum cost. On the one hand, individualmachine degradation should be considered in the machine-levelscheduling. Monitoring techniques of condition-based mainte-nance (CBM) and prognostic & health management (PHM) makeit more feasible to obtain machine degradation data (Lee et al.,2014). On the other hand, modern production characteristicsshould be studied in the system-level scheduling. Therefore, theenabling methods for real-time production and maintenancescheduling are needed to operate in a cost effective manner for

practical applications (Li, Liu, & Chen, 2012; Topal & Ramazan,2010).

As one of modern production patterns, batch production iswidely applied to response quickly to customer demands. This pro-duction pattern has changed the manufacturing process from the‘‘push’’ pattern to the ‘‘pull’’ pattern. In batch production, batchorders are processed through multi-unit systems according to mar-ket demands (Anzanello, Albin, & Chaovalitwongse, 2012;Cardenas-Barron, 2009; Li, 1997). Different from traditional flow-line production, batch production has the following modern pro-duction characteristics: (1) batch orders are independent with var-iable lot size; (2) these sequential batches are ordered only a shorttime beforehand; (3) a set-up work happens when one batchswitches to another; and (4) each batch cycle prefers no interrup-tions to ensure product quality. Therefore, other than classicalopportunistic maintenance, a novel system-level maintenancestrategy is required to eliminate unnecessary production breaks,achieve significant cost reduction and overcome complexity of sys-tem scheduling by considering these production characteristics.

For developing a bi-level maintenance strategy (interactivemachine-level and system-level scheduling), it should be noticedthat batch production systems are usually equipped with variousmachines, which have different reliability parameters and degrad-ing processes. Each machine undergoes increasing wear and

782 T. Xia et al. / European Journal of Operational Research 240 (2015) 781–790

deterioration with aging, which will finally lead to failures andinterrupt the normal production. In the recent decades, numerousvaluable studies have been devoted to single-machine mainte-nance scheduling by combining maintenance plan and productionschedule (Chen, 2006, 2009; Ji, He, & Cheng, 2007; Liao & Chen,2003). In the notable studies, Wang and Liu (2013) developed abranch and bound (B&B) algorithm for production schedulingand preventive maintenance in a single machine with its time tofailure subject to a Weibull probability distribution. Cassady andKutanoglu (2005) proposed an integrated model that coordinatedpreventive maintenance planning decisions with single-machinescheduling decisions so that the total expected weighted comple-tion time of jobs was minimized. Chen (2008) considered the singlemachine scheduling problem and assumed that machine mainte-nance must be undertaken following a given period, where multi-ple periodic maintenance intervals existed, with the maintenanceinterval being flexible through the planning horizon. Comparedwith these groundbreaking studies, the machine-level schedulingin this study focuses on taking advantage of the multiple attributevalue theory and imperfect maintenance to dynamically obtainmaintenance intervals according to machine degradation. Firstly,some other scheduling objectives, besides cost, can be consideredby decision makers (Costa, Carnero, & Oliveira, 2012; Jiang & Ji,2002). The multiple attribute value theory is utilized as a tool toformally handle a multi-attribute decision problem. Secondly, ithas been recognized that PM cannot recover a machine to an ‘‘asgood as new’’ state. Pham and Wang (1996), and more recently,Doyen and Gaudoin (2004) have discussed the maintenance sched-uling with imperfect maintenance. Thirdly, the machine-levelmaintenance intervals should be dynamically integrated into thefollowing system-level scheduling process. The machine-levelscheduling mode is required to support the system-level schedul-ing of preventive maintenance and batch production in a real-timeand sequential mode. Therefore, a multi-attribute model (MAM)developed by Xia, Xi, Zhou, and Du (2012) is used in this study.

Compared with the machine-level scheduling, the system-levelscheduling is much more complex, since the interactions betweenmachines should also be considered in decision making (Dekker,Wildeman, & van der Duyn Schouten, 1997; Okogbaa, Otieno,Peng, & Jain, 2008). With the extensive application of highly com-plex systems, there has been a growing interest in multi-unitmaintenance scheduling. Though the existing complexity of sys-tem scheduling, some valuable system-level maintenance strate-gies have been proposed in the conception of opportunisticmaintenance (Bedford, Dewan, Meilijson, & Zitrou, 2011;Rachaniotis & Pappis, 2008). Opportunistic maintenance basicallyrefers to the scheme in which preventive maintenance is carriedout at opportunities. When one machine in the system fails or ispreventive maintained, PM opportunities arise for other machines.The advantage of opportunistic maintenance is that one PM combi-nation with several PM actions can be used to save much groupmaintenance costs. Zhou, Xi, and Lee (2009) proposed an opportu-nistic PM scheduling algorithm for series systems by maximizingcumulative cost savings. Xia, Xi, Zhou, and Lee (2012) developedthe maintenance time window (MTW) to make a cost-effectiveschedule by dynamically utilizing opportunities in series–parallelsystems. However, this classical type of opportunistic maintenancefor flow-line production focuses on opportunities caused by main-tenance arrangements and does not consider production plans in aholistic manner, thus it is only called maintenance-driven opportu-nistic maintenance. To meet the innovation requirements of batchproduction pattern, there is a great need to develop a novel type ofopportunistic maintenance, which not only considers the degrada-tion of each machine, but also satisfies the characteristics of batchproduction.

In order to propose a production-driven opportunistic mainte-nance strategy suitable for batch production, it is necessary to ana-lyze the impacts of maintenance schedules and batch orders. Thechallenges from production characteristics should be emphasizedin the system-level scheduling. Due to the complexity of jointlyscheduling for preventive maintenance and production plan, thereare very few studies on methodologies for multi-unit degradingsystems (Lee, Wu, Wen, & Chung, 2008; Liao, Shyur, & Lin, 2005;Mellouli, Sadfi, Chu, & Kacem, 2009; Sun & Li, 2010). Allaouia,Lamourib, Artibab, and Aghezzaf (2008) studied the problem ofjointly scheduling n immediately available jobs and the preventivemaintenance in a two-machine flow shop to minimize the make-span. Tan, Chen, and Zhang (2011) considered a parallel-machinescheduling problem with machine maintenance to minimize thetotal completion time. In all, these works play a great role in thejoint scheduling for preventive maintenance and batch production.However, according to batch production characteristics, there stillremain three issues to be addressed. First, most of these strategiesfocus on two-machine scheduling and suffer from intractabilitywhen the number of machines grows. The traditional descriptionof the system condition (i.e., Markov process) makes the analysisextremely complicated in multi-unit system modeling(Wildeman, Dekker, & Smit, 1997). Second, the classical mainte-nance-driven opportunistic maintenance only schedules to advancePM actions. In production practice, it can be extensionally analyzedwhether to advance or to postpone each PM activity. Third, consid-ering the increasing uncertainty of customer demands in batch pro-duction, real-time scheduling ability of multi-unit systems in a costeffective manner is needed in industrial companies that makes var-ious products in large and discrete batches, when demand and theproduction process are stochastic. Therefore, five improvable direc-tions have been considered in the proposed bi-level maintenancestrategy for batch production: (1) dynamic PMs with flexible inter-vals are more practical than periodic maintenance; (2) themachine-level and the system-level scheduling should be interac-tive in a dynamic way; (3) the set-up time between two batchescan be the new opportunities for group maintenance; (4) accordingto the ‘‘pull’’ manufacturing pattern, real-time system-level sched-uling is conducted cycle by cycle; and (5) production-driven oppor-tunistic maintenance based on cost analysis in each cycle canresolve the complexity of system-level scheduling.

In this paper, an interactive bi-level maintenance policy is pro-posed by considering both individual machine degradation andbatch production characteristics. This strategy helps industrialcompanies to eliminate unnecessary production breaks, achievesignificant cost reduction and overcome complexity of systemscheduling. For each single machine, a multi-attribute model(MAM) is used to obtain maintenance intervals based on availabil-ity maximization and cost minimization. Based on real-timemachine-level intervals, an advance-postpone balancing (APB)strategy is presented to operate production-driven opportunisticmaintenance in the system-level scheduling. Each set-up timebetween two batches could be used to perform PM on non-failuremachines, so that unnecessary breakdown in batch productioncould be avoided. The proposed APB programming dynamicallyanalyzes the cost savings of PM advancement and PM postpone-ment, and then makes real-time schedules to satisfy no-disruptionrequirement and reduce system maintenance cost. The rest of thispaper is organized as follows: Section 2 presents the formaldescription of production-driven opportunistic maintenance. Sec-tion 3 illustrates the proposed MAM–APB scheduling method forbatch production with a multi-unit system. In Section 4, case stud-ies are investigated with the proposed methodology and results arediscussed. In Section 5, conclusions are drawn and future workdirections are given.

T. Xia et al. / European Journal of Operational Research 240 (2015) 781–790 783

2. Research design and methodology

For achieving the system cost reduction for batch production,the proposed MAM–APB methodology is applied as the decision-making method, which extends classical opportunistic mainte-nance policies. As an effective method of group maintenance, amaintenance-driven opportunistic maintenance strategy utilizesa PM action of one machine as opportunities for other machines(Xia, Xi, Zhou, & Lee, 2012; Zhou et al., 2009). However, breakdownwithin each batch needs to be avoided to ensure product quality.Thus, each set-up work between batches could be considered asnew opportunities to maintain several machines simultaneously.The scheme of production-driven opportunistic maintenancebased on MAM–APB scheduling is shown in Fig. 1.

This novel production-driven opportunistic maintenance strat-egy consists of four layers: (1) Physical layer: this layer describesand characterizes the complex multi-unit system undertakingbatch production of variable lot size. (2) Data processing layer: pro-duction data flow (order demand, lot size of batch, and productionplan) and maintenance data flow (reliability parameter, machinehazard rate, and maintenance effects) are interactively communi-cated to support the decision-making process. (3) Decision makinglayer: this is the core layer where MAM–APB strategy is applied.According to sequential batches, the real-time PM intervals arederived from the machine-level scheduling process. Then in thesystem-level scheduling, APB programming could dynamically uti-lize the set-up works and analyze the cost savings to make real-time PM adjustments. MAM and APB programs iteratively proceedfrom batch to batch in real time. (4) Application layer: schedulesfrom MAM–APB decision-making process are applied in industrialpractice.

Fig. 1. Scheme of MAM–APB methodology.

Compared with maintenance-driven opportunistic mainte-nance based on maintenance time window (MTW) for flow-lineproduction (Xia, Xi, Zhou, & Lee, 2012), production-driven opportu-nistic maintenance based on APB have following advantages suit-able for batch production. First, APB considers no-disruptionrequirement, other than traditional constant production assump-tion. Second, it takes advantage of planned production downtimesas opportunities. Third, APB is scheduled at each set-up time pointbetween two consecutive batches. Fourth, this strategy dynami-cally analyzes the cost savings of PM advancement and PM post-ponement to make decision. Last but not the least, a set-up workbefore a new batch means a breakdown of all the machines, thusthe complexity of system scheduling can be significantly reducedfor determining whether to advance or postpone PM activities inAPB programming. The detailed scheduling process of MAM–APBmethod will be presented in the following section.

3. MAM–APB strategy for batch production

In the proposed methodology, two kinds of maintenance activ-ities are considered to reduce unplanned downtime in each batchcycle. For each machine, PM is scheduled during set-up timesbetween two batches, while minimal repair is used if it fails duringa batch production. As shown in the decision making layer of Fig. 1,MAM provides PM intervals according to machine degradationcycle by cycle. Given PM intervals and sequential batch orders,APB dynamically makes maintenance schedules by utilizing set-up opportunities. Then real-time APB decisions are fed back toMAM calculation for the next batch cycle. By considering individ-ual machine degradation and batch production characteristics, thisbi-level strategy aims to reduce the total system maintenance cost.

Assumptions considered in this paper are given in thefollowing:

(1) The machines in a batch production system are independentwith different deterioration process. All enter service at timet = 0. The hazard rate at t reflects the health condition.

(2) PM is imperfect maintenance, which does not make themachine be as good as new, but younger. Minimal repaironly recovers the machine to the failure rate it had whenit failed.

(3) Sequential batch orders are independent and can be finishedin corresponding production times. A set-up work ariseswhen one batch switches to another.

(4) The processing time of a single product is negligible in com-parison with the production time of a batch order; hence allmachines can be seen to finish one batch production at thesame time.

Notations

j

index of machine Mj, j 2 {1,2, . . . , J} i index of PM cycles in machine-level scheduling,

i 2 {1,2, . . . , I}

u index of batch Bu, u 2 {1,2, . . . ,U} TBu time duration of batch Bu

tij

the time point of the ith PM at machine-level tbu the set-up time point after Bu at system-level H(j, tbu) maintenance decision for Mj at tbu

Gu

PM combination set after Bu

SCAj(u+1)

saved cost of PM advancement for Mj in Bu+1

SCPj(u+1)

saved cost of PM postponement for Mj in Bu+1

APBj(u+1)

advance-postpone balancing for Mj in Bu+1

csj

set-up cost rate

(continued on next page)

784 T. Xia et al. / European Journal of Operational Research 240 (2015) 781–790

cdj

downtime cost rate kij(t) hazard rate function prior to the ith PM Cpij cost of PM action Tpij time duration of PM action Tpumax maximum duration for PM actions combined in Gu

Cfij

cost of minimal repair Tfij time duration of minimal repair Aij availability of the ith PM cycle crij cost rate of the ith PM cycle Taij PM interval of availability model Tcij PM interval of cost model Toij machine-level PM interval

Fig. 2. Schematic illustration of APB programming.

3.1. Multi-attribute model (MAM) in machine-level scheduling

In the machine-level scheduling, MAM method takes advanta-ges of multiple attribute value theory and imperfect maintenance.In our previous paper (Xia, Xi, Zhou, & Du, 2012), MAM has beenpresented to determine the real-time PM intervals consideringboth machine availability and maintenance cost in an overallobjective function.

In the availability model, a PM cycle consists of two intervals:mean uptime denoted by Taij, and mean downtime. SupposeR Tij

0 kijðtÞdt is the expected frequency of the failures between twosuccessive PM activities. The availability of the ith PM cycle forMj is given by:

Aij ¼Taij

Taij þ Tpij þ T f ijR Taij

0 kijðtÞdt� � ð1Þ

The numerator equals to the useful interval and the denominatorequals to the cycle duration. Output PM interval T�aij correspondingto the maximum A�ij through dAij/dTaijjT = 0.

In the cost model, the cost of PM action and the possible cost ofminimal repair are considered. Suppose Tcij is the PM interval in thecost model. The maintenance cost rate of the ith PM cycle isdefined as:

crij ¼Cpij þ Cf ij

R Tcij0 kijðtÞdt

Tcij þ Tpij þ T f ijR Tcij

0 kijðtÞdt� � ð2Þ

The numerator equals to the maintenance cost and the denomina-tor equals to the cycle duration. Output PM interval T�cij correspond-ing to the minimum c�rij through dcrij/dTcijjT = 0.

Then, MAM method is developed to give an overall objective,denoted by Vij, composed of machine availability and maintenancecost. Suppose w1ij and w2ij are the weights of these two terms(w1ij P 0, w2ij P 0, w1ij + w2ij = 1). By combining two single-objec-tive models together, the overall objective function is thus definedas:

Vij ¼ �w1ijAij

A�ijþw2ij

crij

c�rij

ð3Þ

In this function, PM interval denoted by Toij, takes place of Taij

and Tcij. For the ith cycle for Mj, PM intervals T�oij can be obtainedby dVij/dToijjT = 0. By pulling machine-level PM intervals of allmachines cycle by cycle, the proposed APB programmingdynamically analyzes the cost savings for the system-levelscheduling.

In practice, a machine after PM is not as good as brand new one,that is, the hazard rate value is decreased while always greaterthan zero. Simultaneously, the machine tends to have more fre-quent maintenance since the hazard rate increases more quicklythan it did in the previous PM interval. To sum up, PM not only

decreases the hazard rate to a certain value, but also changes theslope of the hazard rate function. Therefore, for the next PM cycle,given the actual interval Tij from APB feedback, the relationshipbetween hazard rates of consecutive cycles is defined as:

kðiþ1ÞjðtÞ ¼ bijkijðt þ aijTijÞ; t 2 ð0; Tðiþ1ÞjÞ ð4Þ

The age reduction factor aij, aij 2 (0,1) shows that imperfect PMmakes the machine’s initial failure rate reduce to kij(aijTij). Mean-while, the hazard rate increase factor bij > 1 indicates that PMincreases the failure rate bijkij(t). It should be noticed that whent = 0, the hazard rate increase factor has no effect on the failurerate, therefore we could have k(i+1)j(t) = kij(aijTij), t = 0. These factorscan be deduced from the historical maintenance data and onlineoperational state of the machine (Doyen & Gaudoin, 2004; Pham& Wang, 1996).

When each batch production was finished, current machine-level PM intervals would be pulled to APB programming. In thefollowing process, system-level opportunistic maintenancewould be scheduled by taking real-time batch orders intoconsideration.

3.2. Advance-postpone balancing (APB) in system-level scheduling

Consider a multi-unit manufacturing system with J machines,which are of different types and suffer increasing wear at variousrates. For the core system-level scheduling, APB programmingdynamically analyzes the cost savings of PM advancement andPM postponement. This strategy focuses on the key observationthat each PM advancement or postponement incurs three potentialcost savings: downtime cost saving, PM costs saving and minimalrepair cost saving. It aims to make a real-time decision for achiev-ing more cost savings. Thus, unplanned downtime in each batchcould be eliminated and the total system maintenance cost couldbe reduced. The schematic illustration of APB programming isshown in Fig. 2.

The APB program works as follows. When the last batch Bu hasbeen finished, and the next batch Bu+1 has not started, this momentis defined as the current decision time to schedule APB. At this set-up time point tbu, PM opportunities for machines arise and thetime duration of next batch production TBu+1 can be pulled. In thisbatch cycle, PM time points tij for machines M1, Mj and MJ are orig-inally in batch Bu+1. To avoid production breakdown, these PMactions have to be advanced to the current tbu (the current PMcombination set Gu), or postponed to the next tbu+1 (the nextPM combination set Gu+1). However, there is no need to operate

T. Xia et al. / European Journal of Operational Research 240 (2015) 781–790 785

PM twice in the same set-up time; thus the PM of machine MJ

should be postponed.In each batch cycle, APB programming makes real-time decision

of PM advancement or PM postponement by comparing corre-sponding cost savings. On the one hand, if machine Mj is preventedmaintained at time tbu, the saved cost by advancing PM in batchBu+1 can be evaluated as:

SCAjðuþ1Þ ¼ SCAdjðuþ1Þ þ SCAf

jðuþ1Þ � SCApjðuþ1Þ ð5Þ

where SCAdjðuþ1Þ is the downtime cost saving, SCAf

jðuþ1Þ is the minimalrepair cost saving, SCAp

jðuþ1Þ is the PM cost saving of PMadvancement.

Firstly, according to the downtime cost rate cdj and the set-upcost rate csj, the downtime cost saving can be defined as:

SCAdjðuþ1Þ ¼ Tpijðcdj � csjÞ ð6Þ

Secondly, if the PM action is advanced, the PM intervaldecreases to T�oij � ðtij � tbuÞ. Thus the cumulative failure risk isreduced, and the minimal repair cost saving can be represented as:

SCAfjðuþ1Þ ¼

Z T�oij

0kijðtÞdt �

Z T�oij�ðtij�tbuÞ

0kijðtÞdt

" #Cf ij ð7Þ

Thirdly, PM advancement could increase PM cost, since shorterintervals mean that more PM actions would be needed in the samescheduling horizon. According to the ratio of PM interval changeand the actual PM interval, the PM cost saving to be minus canbe shown as:

SCApjðuþ1Þ ¼

tij � tbu

T�oij � ðtij � tbuÞCpij ð8Þ

On the other hand, if PM of machine Mj is postponed to the nextset-up time point tbu+1, the minimal repair cost saving will beminus and the PM cost saving will be plus. Therefore, the savedcost by postponing PM in batch Bu+1 can be evaluated as:

SCPjðuþ1Þ ¼ SCPdjðuþ1Þ � SCPf

jðuþ1Þ þ SCPpjðuþ1Þ ð9Þ

where SCPdjðuþ1Þ is the downtime cost saving, SCPf

jðuþ1Þ is the minimalrepair cost saving, SCPp

jðuþ1Þ is the PM cost saving of PM postpone-ment. Correspondingly, these cost savings can be obtained by thefollowing equations:

SCPdjðuþ1Þ ¼ Tpijðcdj � csjÞ ð10Þ

SCPfjðuþ1Þ ¼

Z T�oijþðtbuþ1�tijÞ

0kijðtÞdt �

Z T�oij

0kijðtÞdt

" #Cf ij ð11Þ

SCPpjðuþ1Þ ¼

tbuþ1 � tij

T�oij þ ðtbuþ1 � tijÞCpij ð12Þ

Based on above cost saving analysis, APBj(u+1) could be definedas the criterion to decide weather to advance or postpone thisPM action. According to the values of SCA and SCP, the APB func-tion can be shown as:

APBjðuþ1Þ ¼ SCAjðuþ1Þ � SCPjðuþ1Þ ð13Þ

If APBj(u+1) > 0, it means that the saved cost of PM advancementis greater, thus PM on machine Mj is advanced to time tbu (j 2 Gu). IfAPBj(u+1) 6 0, the PM action will be postponed to time tbu+1

(j 2 Gu+1). This is the APB decision for one machine in a system-level batch cycle. The procedure of production-driven opportunis-tic maintenance for sequential batches based on MAM–APBscheduling method will be presented in the following section.

3.3. Production-driven opportunistic maintenance based on MAM–APB strategy

To achieve an effective system maintenance schedule for batchproduction, the proposed bi-level strategy not only considers indi-vidual machine degradation, but also integrates batch productioncharacteristics. In essence, APB dynamically determines produc-tion-driven opportunistic maintenances by utilizing set-up worksand analyzing the cost savings, while MAM supplies real-timemachine-level PM intervals. The following procedure obtains sys-tem-level maintenance schedules in sequential batch cycles basedon MAM–APB strategy.

Step 1: Start from the first cycle i = 1. Pull the real-time PMintervals from the machine-level MAM scheduling and evaluateexpected PM time points:

tij ¼ T�oij ðj ¼ 1;2; . . . ; JÞ ð14ÞStep 2: Obtain the first batch order (production duration TB1).Start from the first cycle u = 0, tb0 = 0. Check whether Mj

(j = 1,2, . . . , J) is expected to be maintained in B1:

Hðj; tb0Þ ¼0 tij R ðtb0; tb0 þ TB1�1 tij 2 ðtb0; tb0 þ TB1�

�ð15Þ

Step 3: No PM is needed at tb0 = 0. Thus all PM actions are post-poned to the end of B1. "H(j, tb0) = 1, define APBj1 < 0 andj 2 Gu+1. For u = u + 1 = 1, the second decision time is given by:

tbu ¼ tb0 þ TBu ¼ TB1 ðu ¼ 1Þ ð16Þ

Besides, the expected PM time points of Mj (j = 1,2, . . . , J) have beenadjusted as:

tij ¼tij þ dðGuÞTpu max Hðj; tb0Þ ¼ 0tbu þ dðGuÞTpu max þ T�oijði ¼ iþ 1Þ Hðj; tb0Þ ¼ 1

(

ð17Þ

dðGuÞ ¼0 jGuj ¼ 01 jGuj > 0

�ð18Þ

where jGuj = 0 means there is no PM action in the PM combinationGu. Otherwise, jGuj > 0.

Step 4: Time check: When batch Bu (u = 1,2, . . .) has been fin-ished and a new batch order with TBu+1 is obtained, identifywhether the expected PM actions will be performed duringthe new batch Bu+1:

Hðj; tbuÞ ¼0 tij R ðtbu; tbu þ TBuþ1�1 tij 2 ðtbu; tbu þ TBuþ1�

�ð19Þ

Step 5: APB check: For "H(j, tbu) = 1, analyze the cost savings ofPM adjustments in the system-level scheduling, and use theAPB criterion to advance or postpone PM actions (if we alreadyhave j 2 Gu, define j 2 Gu+1):

j 2Gu APBjðuþ1Þ ¼ SCAjðuþ1Þ � SCPjðuþ1Þ > 0Guþ1 APBjðuþ1Þ ¼ SCAjðuþ1Þ � SCPjðuþ1Þ < 0

�ð20Þ

Step 6: Update time and feedback: For the next batch cycleu = u + 1, update the set-up time point after Bu in the system-level scheduling and the expected PM time points of Mj

(j = 1,2, . . . , J) based on the machine-level scheduling (MAMdetermines new PM intervals according to the actual previousintervals from APB decision through Eq. (4)):

tbu ¼ tbu�1 þ dðGu�1ÞTpðu�1Þmax þ TBu ð21Þ

tij ¼tijþdðGuÞTpumax Hðj;tbu�1Þ ¼0tbu�1þdðGu�1ÞTpðu�1ÞmaxþT�oijði¼ iþ1Þ Hðj;tbu�1Þ ¼1 j2Gu�1

tbuþdðGuÞTpumaxþT�oijði¼ iþ1Þ Hðj;tbu�1Þ ¼1 j2Gu

8><>:

ð22Þ

786 T. Xia et al. / European Journal of Operational Research 240 (2015) 781–790

Step 7: If there is following batch orders coming sequentially,turn to time check for H(j, tbu) in Step 4; perform APB program-ming APBj(u+1) = SCAj(u+1) � SCPj(u+1) to make the system-levelschedules in Step 5; and update time and feedback for the inter-active bi-level scheduling in Step 6. This cyclic system-levelmaintenance scheduling based on APB is shown in Fig. 3.

By applying MAM–APB strategy dynamically, plant managerscan obtain the real-time opportunistic maintenance schedule forbatch production cycle by cycle. To summarize, this interactivebi-level methodology decides the system-level PM arrangementsbased on both production data flow and maintenance data flowwith three steps:

(1) MAM scheduling in the machine-level scheduling: flexible PMintervals considering maintenance effects are scheduledbased on machine degradation. These intervals will supportthe system-level scheduling, and be updated with APB deci-sion feedbacks for the next cycle.

(2) APB programming in the system-level scheduling: based on PMintervals and batch orders, PM actions are scheduled to beadvanced or postponed according to real-time cost savinganalysis. The outputs of the system-level scheduling includePM combinations Gu, set-up time points tbu and breakdowndurations for PM Tpu max.

(3) System-level performanceevaluation: the total system mainte-nance cost is evaluated by system-level outputs. Let csj bethe set-up cost rate, Cpij be the cost of PM action, and Cfij

be the cost of minimal repair. The total system maintenancecost with U batches can be evaluated by:

ETC ¼XU�1

u¼0

dðGuÞTpu max

XJ

j¼1

csj

!þXI

i¼1

Xj2Gu

Cpij

þXI

i¼1

XJ

j¼1

Cfij

Z TUpdatedoij

0kijðtÞdt

!ð23Þ

Fig. 3. Flowchart of system-leve

In Eq. (23), it can be found that the total system maintenancecost consists of three cost parts: the first part is the total set-upcost for sequential batches; the second part is the total PM costfor all machines; the third part is the total minimal repair cost insequential maintenance cycles. It should be noticed that TUpdated

oij

means the actual PM intervals according to APB programming:PM advancement makes TUpdated

oij ¼ T�oij � ðtij � tbuÞ, while PM post-ponement makes TUpdated

oij ¼ T�oij þ ðtbuþ1 � tijÞ.Moreover, traditional maintenance-driven opportunistic main-

tenance policies calculate the cost-savings of all possible PM com-binations at every decision time. With the number of machinesgrows, possible PM combinations will increase exponentially,which means the system-level scheduling will suffer from intracta-ble complexity. As a dynamic method, APB can provide decisionoutputs at each set-up time point according to real-time cost sav-ing analysis. Therefore, production-driven opportunistic mainte-nance based on MAM–APB methodology can effectively handle amanufacturing system with growing machine number and change-able batch orders.

4. Numerical results and discussion

To validate MAM–APB methodology, a complex manufacturingsystem consists of seven various machines is considered here, asshown in Fig. 1. For dynamically scheduling production-drivenopportunistic maintenance, both maintenance data flow and pro-duction data flow are synthetically collected in a hydraulic steeringfactory.

In the maintenance data flow, the reliability of each machineis formulated by a Weibull failure probability functionk1jðtÞ ¼ ðmj=gjÞðt=gjÞ

mj�1, which has been widely used to fit repair-able equipment in electronic and mechanical engineering. Relativeparameters are estimated by maintenance engineers and pre-sented in Table 1.

In the production data flow, batch orders are processed throughthis 7-unit system based on fast-paced market demands. Accordingto ‘‘pull’’ production pattern, new batch order would be placed

l scheduling based on APB.

Fig. 4. Production data of batches.

T. Xia et al. / European Journal of Operational Research 240 (2015) 781–790 787

only at each decision time. The lot sizes of sequential batches aregiven in Fig. 4.

4.1. MAM results in machine level

In the machine-level scheduling, MAM takes advantages of mul-tiple attribute value theory and imperfect maintenance to findmaintenance requirements according to individual machine degra-dation. This study considers the situation that maintenanceresources are available in the machine-level scheduling, whilethe system-level maintenance opportunities are caused by batchproduction characteristics. The machine-level outputs of PM inter-vals would be provided for the system-level scheduling at each set-up time point.

As concluded by Xia, Xi, Zhou, and Du (2012), these machine-level results reveal following conclusions: (1) the PM intervaldecreases while PM cycle increases, since the underlying hazardrate evolution becomes faster with the degradation process; (2)machine availability will be lower and maintenance cost will behigher as a machine ages due to the consideration of maintenanceeffects; and (3) ignoring the effects of a maintenance activity leadsto less availability and extra cost, and MAM contributes to morepracticality of PM intervals.

4.2. APB scheduling in system level

Our focus is on the effectiveness of the system-level schedulingfor batch production. PM intervals from the machine-level sched-uling usually lead to unnecessary production breakdown. Thus, ateach set-up time point, APB strategy dynamically analyzes the costsavings of PM advancement and PM postponement, and thenmakes real-time schedules to satisfy no-disruption requirement.APB programming at the third set-up time point is taken as anexample. Table 2 shows the scheduling results of expected PMactions in batch 4.

It can be seen that when batch 3 has been finished (tb3 = 8700),PM opportunities appear and the time duration TB4 = 5000 isordered. At this decision time, the expected PM time points tij forM1, M2, M3, M5, M6 and M7 are originally scheduled to be oper-ated in B4.

For example, according to the machine-level scheduling, PM ofM1 was arranged to be performed at time 12,129. However, thisPM action (original interval T�o21 ¼ 5829) should be advanced(TA

o21 ¼ 2400, uptime of TB3) or postponed (TPo21 ¼ 7400, uptime

of TB3 and TB4) in the system-level scheduling. Since APB14 = �3269is negative, PM of M1 should be postponed to G4. Other calculatedAPB values at this set-up time point are given in Table 3.

It can be seen that M1’s downtime cost can be reduced asSCAd

14 ¼ SCPd14 = 20,000, since PM is moved out of batch 4. If PM is

advanced, the minimal repair cost SCAf14 ¼ 3862 can be saved,

while the PM cost SCAp14 ¼ 9287 will be increased. In sum, the

saved cost of PM advancement is SCA14 = 14,575, while PM post-ponement leads to SCP14 = 17,844. Therefore, APB14 = �3269 meansPM postponement can save more maintenance cost. From Tables 2

Table 1Maintenance data of machines.

Mj (mj, gj) (aij, bij) Tpij Tfij Cpij Cfij cdj csj

M1 (2.5, 10,000) (0.04, 1.05) 200 600 6500 15,000 120 20M2 (1.5, 8000) (0.02, 1.035) 80 400 3000 7500 70 20M3 (3, 12,000) (0.03, 1.02) 150 700 4000 8000 60 20M4 (2.8, 11,000) (0.025, 1.06) 240 450 8000 10,000 100 20M5 (1.8, 7000) (0.04, 1.04) 100 300 5000 12,000 65 20M6 (2.4, 13,000) (0.035, 1.03) 200 360 2000 6000 75 20M7 (3.2, 15,000) (0.05, 1.025) 300 800 8500 18,000 140 20

and 3, the system-level PM arrangements at the third set-up timepoint could be obtained, and the same APB programming would beperformed at other sequential set-up time points.

4.3. Results of production-driven opportunistic maintenance

Faced with sequential batch orders, APB strategy dynamicallyutilizes set-up works and analyzes the cost savings to reduce thetotal system maintenance cost. PM intervals from the machinelevel and various batch orders are pulled to make opportunisticmaintenances cycle by cycle. And the APB decision feedback willbe returned to update PM intervals for the next cycle. The resultsof production-driven opportunistic maintenance are presented inTables 4 and 5.

In Table 4, a positive value indicates that the saved cost of PMadvancement is greater, the machine should be maintained inadvance, while a negative value makes PM to be postponed. Itcan be found that a larger Cfij promotes PM advancements toreduce cumulative failure risks, while a larger Cpij leads to PM post-ponements to avoid extra maintenance activities. In summary, APBchooses every PM adjustment with a large cost saving for everymachine at each set-up time point. Correspondingly, Table 5 showsthe complete layout of these opportunistic maintenances.

4.4. Effectiveness of MAM–APB methodology

To validate the proposed bi-level methodology of production-driven opportunistic maintenance for batch production, we inves-tigate the cumulative cost-saving achieved by APB programming ineach batch. Furthermore, we compare the expected total systemmaintenance cost (ETC) of MAM–APB scheduling with other oppor-tunistic maintenance policies to show the significant cost reduc-tion. The cumulative cost-savings achieved in sequential batchesare presented in Fig. 5.

Furthermore, three maintenance-driven opportunistic mainte-nance policies discussed by Xia, Xi, Zhou, and Lee (2012) are usedto make a comparison with APB strategy in the horizon of280,000 h: (1) individual maintenance mode (IMM): PM is con-ducted on a machine only when it reaches its original PM intervals;(2) simultaneous maintenance mode (SMM): when one of themachines reaches its intervals, PM actions are carried out on allmachines; and (3) maintenance time window (MTW): onemachine’s PM arises PM opportunities of non-failed machineswithin that batch.

Besides, two traditional production-driven opportunistic main-tenance policies are introduced in the comparison: (1) advancedmaintenance mode (AMM): PM actions originally planned to beperformed in the next batch are all shifted to the current set-up

Table 2APB programming at the third set-up time point.

Mj tb3 TB4 tij H(j, tb3) TAoij

T�oij TPoij

APBj4 j 2 G3 j 2 G4

M1 8700 5000 12,129 1 2400 5829 7400 �3269 YM2 12,994 1 2400 6694 7400 33 YM3 13,664 1 2400 7364 7400 �6332 YM4 16,638 0 – – – – – –M5 10,985 1 2400 4685 7400 5762 YM6 13,221 1 2400 6921 7400 �2332 YM7 9474 1 8500 9274 13,500 6490 Y

Table 3Cost saving analysis at the third set-up time point.

j SCA SCAj4 SCP SCPj4 APBj4

SCAdj4 SCAf

j4SCAp

j4 SCPdj4 SCPf

j4 SCPpj4

1 20,000 3862 9287 14,575 20,000 3516 1380 17,844 �32692 4000 4712 5367 3345 4000 974 286 3312 333 6000 1931 8273 �342 6000 29 19 5990 �63324 – – – – – – – – –5 4500 4376 4761 4115 4500 7982 1835 �1647 57626 11,000 1316 3768 8548 11,000 249 129 10,880 �23327 36,000 940 774 36,166 36,000 8984 2660 29,676 6490

Table 4APB results in sequential batch cycles.

APB (cost) B1 B2 B3 B4 B5 B6 B7 B8 B9 B10

M1 �9204 �3269 �2725 4022M2 2262 33 1865 1746 2790M3 392 �6332 105 �902M4 �110 4033 4555M5 �78 5762 5934 3687 1758 �1640M6 526 �2332 336 151M7 6490 1951 2250

Table 5Results of production-driven opportunistic maintenance.

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10

Schedule (time point) tb1 tb2 tb3 tb4 tb5 tb6 tb7 tb8 tb9 tb102000 6100 8700 14,000 16,500 17,000 20,300 22,200 24,700 28,300

M1 Y Y Y Y –M2 Y Y Y Y Y –M3 Y Y Y YM4 Y Y Y –M5 Y Y Y Y Y Y –M6 Y Y Y Y –M7 Y Y Y –

788 T. Xia et al. / European Journal of Operational Research 240 (2015) 781–790

time point; (2) postponed maintenance mode (PMM): all the PMactivities are delayed to the end of that batch. The total systemmaintenance costs of APB programming and above five policiesare shown in Fig. 6.

Based on the results in Fig. 6, it can be found that ETC of MTWpolicy is 1,273,728. This is the lowest in maintenance-drivenopportunistic maintenance policies, and MTW has been provenas a cost-effective strategy (ETC-saving rate 46.06% with IMMand 10.82% with SMM). However, huge downtime cost caused byproduction interruptions increases ETC value. Thus, production-driven opportunistic maintenance policies are more suitable forbatch production. Here we can see that ETC of APB policy is487,542, which is the most cost-effective system-level strategy(ETC-saving rate 16.71% with PMM and 10.86% with AMM).

In the general sense, MAM–APB methodology can efficientlyeliminate unnecessary production breaks, achieve significant cost

reduction and overcome complexity of system scheduling withvarious maintenance data flow and production data flow. Differentmanufacturing systems with various machine reliabilities andchangeable batch orders would lead to different ETC-saving rates.However, the mechanism of APB programming can ensure thedynamic scheduling performance. On the one hand, with sequen-tial PM advancement or postponement, this production-drivenopportunistic maintenance strategy eliminates unnecessary pro-duction breaks by utilizing set-up opportunities between succes-sive batches. Huge downtime cost saving ensures that ETC of APBpolicy is much lower than those of maintenance-driven opportu-nistic maintenance policies (e.g. IMM, SMM and MTW). On theother hand, APB programming dynamically compares cost savingsand chooses the PM adjustment with MaxfSCAju; SCPjug at each set-up opportunity. This sequential decision-making process ensuresthat the maximization of ETC-saving rate can be achieved, which

Fig. 5. Cumulative cost-saving in sequential batches.

Fig. 6. Results comparison of opportunistic maintenance policies.

T. Xia et al. / European Journal of Operational Research 240 (2015) 781–790 789

makes APB more effective than traditional production-drivenopportunistic maintenance policies (e.g. AMM and PMM). In sum,APB method achieves significant system cost reduction because itnot only considers batch production characteristics but also makesthe PM adjustment according to the larger cost saving for everymachine at each set-up time point.

5. Conclusions

In this paper, a production-driven opportunistic maintenancepolicy for batch production is proposed based on MAM–APB sched-uling method. This bi-level maintenance strategy systematicallyconsiders not only individual machine degradation, but also batchproduction characteristics. Other than classical maintenance-dri-ven opportunistic maintenance, the new methodology utilizesset-up opportunities to eliminate unnecessary production breaks,achieve significant cost reduction and overcome complexity of sys-tem scheduling. The developed MAM method is used to obtainmaintenance intervals based on availability maximization and costminimization. Furthermore, APB strategy dynamically analyzes thecost savings of PM advancement and PM postponement, and thenmakes real-time schedules to satisfy no-disruption requirementand reduce system maintenance cost. The cost savings achievedby applying this bi-level maintenance strategy have been demon-strated through a case study of seven-unit system. Results indicatethat the total system maintenance cost saving achieved by APBstrategy is much higher than traditional opportunistic mainte-nance policies. It can be concluded that proposed MAM–APB meth-odology is a viable and effective policy to reduce systemmaintenance cost for batch production.

Further research is required for improving the industrial imple-mentation and demonstration of the newly proposed APB policy.Although this production-driven opportunistic maintenance cansignificantly reduce the system maintenance cost, how to rapidlyreact to practical changes and updates of batch orders needs tobe studied, especially in the global competitive market. Further-more, how to introduce the limitation of maintenance resourcesinto the MAM–APB strategy will be investigated in future studies.

Acknowledgements

The authors would like to thank the editor and three anony-mous referees for their remarkable comments. The research isfunded partially by Programme of Introducing Talents of Disciplineto Universities (B06012), Foundation for Innovative ResearchGroups of the National Natural Science Foundation of China(50821003), National Natural Science Foundation of China(51075277, 51275558), Program for New Century Excellent Talentsin University (NCET-11-0328) and National Science FoundationAward # 0825789 at the NSF Sponsored I/UCRC Center forIntelligent Maintenance Systems at the University of Michigan.

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